Existence And Global Attractivity Of Periodic Solution For Neutral Functional Differential Equations

By applying Mawhin's continuation theorem(MCT) and Lyapunov's second method, this thesis mainly studies the existence and global attractivity of periodic solution for several types of neutral functional differential equations(NFDEs), and investigates the existence and uniqueness of positive periodic solution for a Volterra model with mutual interference.In the first Chapter, we introduce the history of functional differential equation and the work of this paper. Chapter 2 introduces some necessary lemmas and establishes several inequalities. In Chapter 3, by utilizing MCT we analysis two kinds of higher order NFDEs with distributed delay. Differing from the previous results we mainly study the influence of distributed delay which is more realistic than discrete delays, and we generate the results to higher order equations. In Chapter 4, we study the existence of globally attractive periodic solution for neutral differential system(NDS) with delays. As far as we know there are some papers on the existence of periodic solutions for NDS, but the methods they used are the exponential dichotomy and Krasnoselakii fixed point theorem, and the sufficient conditions for the existence of periodic solutions which they obtained are under the condition that the difference operator D is stable(i.e.|c| < 1). The main reason why few scholars have studied the global attractivity of periodic solution for NDS is that the general way is to construct a suitable Lyapunov function. It is regrettable that there is no general methods to construct the suitable Lyapunov function, so how to construct a suitable Lyapunov function is a difficult work. By using MCT we relax the conditions which guarantee the existence of periodic solutions to that the operator D is unstable (i.e.|c| > 1), under the stability of the operator D and by constructing a suitable Lyapunov function we obtain some sufficient conditions for insuring the existence of only one globally attractive periodic solution. Lastly, we discuss the existence and global attractivity of periodic solution to a kind of NFDE with multiple deviating arguments. There are few works on the global attractivity of periodic solution for NFDEs. Our work improves the previous results. We mainly investigate the existence and uniqueness of positive periodic solution for a Volterra model with mutual interference in the last Chapter.